A Fine-Grained Analysis of Millimeter-Wave Device-to-Device Networks

Transcription

1 A Fine-Grained Analysis of Millimeter-Wave Device-to-Device Networks Na Deng and Martin Haenggi, Fellow, IEEE Abstract Enabling device-to-device DD communications in millimeter-wave mm-wave networks is of critical imortance for the next-generation mobile networks to suort very high data rates multi-gigabits-er-second for mobile devices. In this aer, we rovide a fine-grained erformance analysis of the mm-wave DD communication networks. Secifically, we first establish a general and tractable framework to investigate the erformance of mm-wave DD networks using the Poisson biolar model integrated with several features of the mm-wave band. To show what fraction of users in the network achieve a target reliability if the required signal-to-interference-lus-noise ratio SINR or QoS requirement is given, we derive the meta distributions of the SINR and the data rate. Interestingly, in mm-wave DD networks, the standard beta aroximation for the meta distribution does not work very well when highly directional antenna arrays are used or the node density is small. To resolve this issue, we rovide a modified aroximation by using higher moments of the conditional SINR distribution, which is shown to be closer to the exact result. On this basis, we also derive the mean local delay and satial outage caacity to rovide a comrehensive investigation on the imact of mm-wave features on the erformance of DD communication. Index Terms Stochastic geometry; Poisson oint rocess; meta distribution; mean local delay; satial outage caacity; millimeter wave; DD communication. A. Motivation I. INTRODUCTION With the exlosive growth of mobile devices and emerging alications, the tension between caacity requirements and sectrum shortage becomes increasingly rominent. To mitigate this dilemma, the two otions are to exloit new sectrum resources or to increase the sectrum efficiency. Recently, the wireless industry has turned its attention to the millimeter wave mm-wave band, between 3 and 3 GHz, execting to take advantage of the huge and unexloited bandwidth to coe with the future multi-gigabit-er-second data rate demand [, 3]. Besides using new wireless sectrum, 5G systems will also exloit intelligence at the device side e.g., by allowing deviceto-device DD connectivity to rovide higher sectrum efficiency, reduced end-to-end latency and increased throughut [3]. Generally, DD connections can be set u between Na Deng is with the School of Information and Communication Engineering, Dalian University of Technology DLUT, Dalian, 64, China dengna@dlut.edu.cn. Martin Haenggi is with the Det. of Electrical Engineering, University of Notre Dame, Notre Dame 46556, USA mhaenggi@nd.edu. Part of this work is under submission at the 7 IEEE Global Communications Conference GLOBECOM 7 []. This work was suorted by the Fundamental Research Funds for the Central Universities DUT6RC39 and by the US NSF grant CCF two devices either directly or by relaying with or without base station BS involvement. For existing communication systems, the main challenge of imlementing underlaid DD links is the difficulty in interference management between the intra-dd interference and the cross-tier interference from other systems using the same sectrum, esecially for the autonomous case without BS involvement [4]. However, since the mm-wave sectrum has several unique features such as sectrum availability, high roagation loss, directivity, and sensitivity to blockage, the situation will be different when the DD communication occurs in the mm-wave band. For examle, since the antenna dimension is inversely roortional to the frequency, the mm-wave DD users equied with very small and highly directional antennas cause much less interference than the DD users adoting omni-directional antennas. Another imortant benefit is that the new sectrum facilitates the more wide-sread use of autonomous DD communication. By oerating in mm-wave band, there is no cross-tier interference between the DD network and the sub- 6 GHz cellular systems, esecially for the autonomous tye where the cross-tier interference is quite serious and hard to be controlled if both DD and cellular transmissions occur in the same band. Accordingly, both the signaling overheads and the loads of BSs can be reduced significantly. Thus, mm-wave DD communication is of critical imortance for the next-generation mobile networks and deserves in-deth investigations on how to exloit the salient roerties of mmwave communication for DD networks and to analyze their erformance. This is the focus of our aer. B. Related Work Stochastic geometry has been successfully alied to model and analyze wireless networks in the last two decades since it not only catures the toological randomness in the network geometry but also leads to tractable analytical results [5 7]. Prior work on mm-wave based networks has mostly used the Poisson oint rocess PPP to model the satial distribution of nodes and analyzed coverage and rate while modeling the directionality of antennas and the effect of blockages [8 ]. For DD-based networks, there also exists a growing body of literature using the PPP to model the irregular satial structure of DD users and base stations [, ]. Very recently, researchers have started to ay attention to the benefits of combining mm-wave and DD technologies. Secifically, [3] studied the ath loss behavior in urban environment for mmwave DD links based on a ray tracing simulation. [4] considered DD connectivity in mm-wave networks, where

2 the robability distribution, mean, and variance of the interdevice distance were derived, and the connectivity erformances of both the direct and indirect DD communications were investigated. [5] used stochastic geometry to analyze the erformance of mm-wave DD networks with a finite number of interferers in a finite network region. These works investigated the mm-wave DD networks either through ure simulations or theoretical analysis with a finite satial extent. Only [6] analyzed the downlink coverage robability of a DD relay-assisted mm-wave cellular network where the obstacles, BSs, and users formed indeendent homogeneous PPPs. The signal-to-interference-lus-noise ratio SINR is a fundamental metric to understand how a communication system/network erforms. When stochastic geometry is used for the analysis, the SINR erformance is most commonly characterized as the success robability relative to an SINR threshold and evaluated at the tyical link. However, the erformance of the tyical link reresents an average over all the satial realizations of the oint rocess or over all the links in a single realization if the oint rocess is ergodic, which rovides very limited information on the individual links. To overcome this limitation, [7] introduced the concet of the meta distribution, which is the distribution of the conditional success robability given the oint rocess. Conditioned on the PPP, the success robability of each individual link is calculated by averaging over the fading and the random activity of the interferers. The distribution of these conditional success robabilities, obtained by an exectation over the oint rocess is the meta distribution. In contrast, the standard success robability is the mean of the conditional success robability. Consequently, the meta distribution rovides a much sharer characterization of the network erformance i.e., the SINR erformance. While [7] focused on the meta distribution of the signal-to-interference ratio SIR in both the Poisson biolar networks with ALOHA channel access and the downlink of Poisson cellular networks, [8] and [9] alied the SIR meta distribution to study the DD communication underlaying cellular networks and fractional ower control for cellular networks in microwave bands, resectively. Different from them, we focus on the meta distributions of the SINR and the achievable rate in mm-wave DD networks considering the unique channel characteristics and antenna features of mmwave communications. C. Contributions The main objective of this aer is to introduce and romote the meta distribution as a key erformance metric for mmwave DD networks. The desirable features of mm-wave, which include sectrum availability, small antenna dimensions enabling the imlementation of highly directive antenna arrays, natural interference suression, and dense deloyability, motivate us to carry out a comrehensive investigation on the erformance of the network by combining the two romising technologies, i.e., mm-wave and DD, aiming at finding the most efficient way to oerate DD in mm-wave frequencies. Secifically, we first give a closed-form exression for the moments of the conditional success robability for mm-wave DD networks considering the effects of blockage and the large beamforming gain from directional antenna arrays. Next, to show what fraction of users in the network achieve a target reliability or transmission effectiveness if the SINR or QoS requirement is given, we rovide analytical exressions for the exact meta distributions of the SINR and the data rate, which are the distributions of the conditional success robability and the conditional achievable rate given the oint rocess, resectively. Due to the unique features of mm-wave, the standard beta distributed aroximation roosed for microwave networks does not work very well when highly directional antenna arrays are used or the network node density is small. Thus, we roose a general beta distribution as a modified and more accurate aroximation for the meta distribution. On this basis, the mean local delay and satial outage caacity SOC first introduced in [] are also calculated for mm-wave DD links and networks, resectively. Finally, the imacts of mmwave features, the link distance between DD users, and the density of users on each erformance metric are investigated numerically, which show that the unique features of the mmwave band and the user density have significant effects on the interference i.e., when the network is interference-limited and when it is not and hence the erformance of mm-wave DD networks. D. Organization The rest of the aer is organized as follows: Section II introduces the system model with blockage effect and antenna attern gain. Section III gives a general framework for a fine-grained analysis of mm-wave DD networks, including the moments of the conditional success robability, the meta distributions of the SINR and the data rate, the mean local delay, and the satial outage caacity. Section IV resents the numerical results, and Section V offers the concluding remarks. A. Network Model II. SYSTEM MODEL We consider a mm-wave DD communication network, where the DD transmitters are distributed according to a homogeneous PPP Φ with density λ. Each transmitter is assumed to have a dedicated receiver at distance r in a random orientation, i.e., the DD users form a Poisson biolar network [7, Def. 5.8]. We consider a receiver at the origin that attemts to receive from an additional transmitter located at r,. Due to Slivnyak s theorem [7, Thm. 8.], this receiver becomes the tyical receiver under exectation over the PPP. We assume that each receiver has a single antenna and its corresonding transmitter is equied with a square antenna array comosed of N elements. All transmitters oerate at a constant ower µ and aly analog beamforming to overcome the severe ath loss in the mm-wave band. We also assume that the direction of arrival DoA between the transmitterreceiver air is known at the transmitter and thus the beam direction is erfectly aligned to obtain the maximum ower gain. The ALOHA channel access scheme is adoted, i.e., in each time slot, DD transmitters in Φ indeendently transmit with robability.

3 3 TABLE I. Antenna arameters of a uniform lanar square antenna array [6] Parameters Descrition value w Half-ower beamwidth 3 N G m Main lobe gain N G s Side lobe gain / sin 3π N B. Blockage and Proagation Model The generalized LOS ball model [] is used to cature the blockage effect in mm-wave communication since it has been validated in [9] as a better fit with real-world scenarios than other blockage models adoted in revious works. Secifically, the LOS robability of the channel between two nodes with searation d in this model is P LOS d = d < R, where is the indicator function, R is the maximum length of a LOS channel, and the LOS fraction constant [, ] reresents the average fraction of the LOS area within a circular ball of radius R around the receiver under consideration. Thus, is the LOS robability if the distance d is less than R. The blockage effect induces different ath loss exonents, denoted as α L and α N, for LOS and NLOS channels, resectively. Tyical values for mm-wave ath loss exonents can be found in measurement results in [] with aroximated ranges of α L [.9,.5] and α N [.5, 4.7]. For the sake of mathematical tractability, the sectorized antenna model is adoted to aroximate the actual antenna attern, as in []. In articular, the array gains within the half-ower beam width are assumed to be the maximum ower gain i.e., main lobe gain, and the gains of the other DoAs are aroximated to be the first minor maximum gain i.e., side lobe gain of the actual antenna attern, which can be formulated as { Gm if φ w/ Gφ = otherwise, G s where w, π] is the half-ower beam width and correlated with the size of antenna array, and φ [ π, π is the angle off the boresight direction. With the assumtion of a N N uniform lanar square antenna array with half-wavelength antenna sacing, the half-ower beamwidth w, main lobe gain G m, and side lobe gain G s are summarized in Table I. C. SINR Analysis We assume that the desired link between the transmitterreceiver air is in the LOS condition with deterministic ath loss r α L. In fact, if the receiver was associated with a NLOS transmitter, the link would quite likely be in outage due to the severe roagation loss and high noise ower at mm-wave bands as well as the fact that the interferers can be arbitrarily close to the receiver. The ower fading coefficient associated with node x Φ is denoted by h x, which is an exonential random variable with Eh x = Rayleigh fading for both LOS and NLOS to enhance the analytical tractability, and all h x are mutually indeendent and also indeendent of the oint rocess Φ. lx is the random ath loss function associated with the interfering transmitter location x, given by { x α L w.. P lx = LOS x x α 3 N w.. P LOS x, where all lx x Φ are indeendent. For the tyical receiver, the interferers outside the LOS ball are NLOS and thus can be ignored due to the severe ath loss over the large distance at least R. As a result, the analysis for the network originally comosed by the PPP Φ with density λ reduces to the analysis of a finite network region, namely the disk of radius R centered at the origin. Due to the incororation of the blockages, the LOS transmitters with LOS roagation to the tyical receiver form a PPP Φ L with density λ, while Φ N with density N λ is the transmitter set with NLOS roagation, where + N = such that Φ = Φ L Φ N. Thus, the interference at the origin is defined as I x Φ µgφ x h x lxb x, 4 where µ is the constant transmit ower, Gφ x is the directional antenna gain function with DoA φ x, and B x is a Bernoulli variable with arameter to indicate whether x transmits a message to its receiver. Since all transmitters are oriented toward the corresonding receivers, the DoAs between the interferers and the tyical receiver are uniformly random in [ π, π. Hence, Gφ x is equal to G m with robability w = w π and G s with robability w. Without loss of generality, the noise ower is set to one, and the SINR at the tyical receiver is then given by µg m h x r α L SINR = +. 5 µgφ x h x lxb x x Φ III. A GENERAL FRAMEWORK FOR FINE-GRAINED ANALYSIS OF MM-WAVE DD NETWORKS In this section, we will develo a general framework for the fine-grained analysis of mm-wave DD networks. The main results include the exact analytical exression and a general beta aroximation for the meta distribution of the tyical DD receiver, which will then be alied in the analysis of the local delay and the SOC for mm-wave DD networks. A. Meta Distribution of the SINR The meta distribution of the SINR is a two-arameter distribution function defined as [7] F Psθx P! op s θ > x, θ R +, x [, ]. 6 Note that the LOS mm-wave links are better modeled by the Nakagami fading. However, we resort to Rayleigh fading as it enables much better tractability. In addition, simulation results in Sec. IV show that Nakagami and Rayleigh fading resent the same trends in terms of SINR erformance. With the standard ath loss law r α, for α, the interference is infinite almost surely unless the network considered is finite, while for α >, the interference is finite but with infinite mean due to the singularity of the ath loss law at the origin [3]. Accordingly, in our system model, we assume α N > maxα L,.

4 4 which reresents the comlementary cumulative distribution function CCDF of the link success robability P s θ conditioned on the oint rocess. Here P! o denotes the reduced Palm robability conditioning on the tyical receiver at the origin o and its corresonding transmitter to be active, and the link success robability P s θ is a random variable given as P s θ PSINR > θ Φ, 7 where θ is the SINR threshold. Due to the ergodicity of the oint rocess, the meta distribution can be interreted as the fraction of links in each realization of the oint rocess that have a SINR greater than θ with robability at least x. By such a definition, the standard success robability is the mean of P s θ, obtained by integrating the meta distribution 6 over x [, ]. Since a direct calculation of the meta distribution seems infeasible, we will derive the exact analytical exression through the moments M b θ E [ P s θ b] first and then aroximate it with much simler closed-form exressions. Theorem. Moments for mm-wave DD network with ALOHA Given that the tyical link is LOS and active, the moment M b b C of the conditional success robability in mm-wave DD networks is M b θ=ex λπr A L N A N bθrα L µg m, 8 where b n n A i = n w m w n m F m,n m α i, θ, 9 n m and n= m= F m,n m α i, θ= F δ i, m, n m, δ i +, Rα i, G mr α i G s θr α. L Here F is the hyergeometric function of two variables 3 [4, Cha. 9.8] and δ i = /α i, i {L, N}. Proof: See Aendix A. The first moment of the conditional success robability is the standard success robability for the mm-wave DD network, denoted as s θ or M θ. It is given as s θ=m θ=ex λπr ξ L + N ξ N θrα L where θr α + w L F, δ i, δ i +, G mr α i G s θr αl ξ i = w F, δ i, δ i +, Rα i R µg m,, i {L, N}. When α L = and α N = 4, we have the simler exressions ξ L = w θr + R ln R θr + w G sθr + G m R ln G mr G s θr, ξ N θr R = w arctan Gs θr + w θr Gm R arctan Gm R. θgs r 3 The F function is also called the Aell function and is imlemented in the Wolfram Language as AellF[a, b, b, c, x, y]. The second moment of the conditional success robability is given as M θ = M θ ex λ πr ξ L + N ξ N, 3 where ξ i = w F, δ i, δ i +, Rαi θr αl w F, δ i, δ i +, G mr α i G s When α L = and α N = 4, we have + w wf, α i, θ +, i {L, N}. 4 ξ L = w θr θr + + w G s θr R G s θr + G mr + w wg sθr G m G s R ln Gs θr +G m R G s θr +G sr, ξ N = w θr R arctan R + θr θr θr +R4 w Gs θr + R arctan R + θr θr θr +R4 + w wg s θr Gm R G m G s R arctan arctan R. Gs θr θr The variance of the conditional success robability can be obtained as var P s θ = M θ M θ. As in [7], we also find in mm-wave DD networks that the same λ leads to the same standard success robability but the variance deends on both λ and, not just the roduct, which highlights the imortance of the fine-grained analysis based on the meta distribution. Moreover, in mm-wave DD networks, as in biolar networks with standard ath loss and single-antenna nodes, we also have the following concentration roerty. Corollary. Concentration as Keeing the transmitter density t λ fixed and letting, we have lim P sθ = M θ = s θ 5,λ=t in mean square and robability and distribution. Proof: From the exression of ξ i in 7, when, we have ξ i = ξ i and thus M θ = M θ. In this case, the limiting variance is zero, i.e., lim var P sθ =. 6,λ=t The concentration roerty means that in the limit of a very dense network with very small, all links in the network have exactly the same success robability or reliability. For mm-wave networks, it is very imortant and interesting to exlore how the erformance changes with the antenna array size N and what haens when extremely massive antenna arrays are used, i.e., as N. Both questions are answered in the following corollary. Corollary. Monotonicity with N For b R +, M b is monotonically increasing with N. When N, M b is given by lim M bθ = ex λπr A L + N A N, b C, 7 N

5 5 where A i = b n+ n F n n= n, δ i, δ i +, 9π R α i 4. 8 Here F is the Gaussian hyergeometric function [4, Cha. 9.], and δ i = /α i, i {L, N}. Proof: See Aendix B. So when N, we obtain an uer limit of the standard success robability M θ as lim M θ = N ex λπr i F, δ i, δ i +, 9π R α i. 4θr α 9 L This corollary imlies that the standard success robability imroves with increasing N and converges to its maximum as N. Furthermore, the limit 7 also rovides the following insights: As N, the effect of the noise is totally suressed since G m ; The side lobe interference gradually becomes the main erformance-limiting factor as N. In this regime, the node density lays a critical role in determining the interference ower in the network. For instance, a sarse network makes the interference in a massive MIMO network arbitrarily small while a dense one is interference-limited. The exact meta distribution can be obtained by the Gil- Pelaez theorem [5] with the imaginary moments M jt of P s θ, t R, j. Corollary 3. Exact exression The meta distribution for the mm-wave DD networks is given by e λπr Rζ F Psθx = π t sin t log x+t θrα L λπr Iζ dt, µg m where ζ = A L + N A N is given in 9 with b = jt and Rz and Iz denote the real and imaginary arts of z C, resectively. Proof: According to the Gil-Pelaez theorem, the ccdf of P s θ is given by F Ps θx = + I e jt log x M jt dt, π t where M jt, t R is given in 8, and Iz is the imaginary art of z C. Letting ζ = A L + N A N, we have F Ps θx = + π = + π = π I e jt log x e λπr ζ jt /µg m t dt e λπr Rζ I α L µgm dt e jt log x e jλπr Iζ jt θr t e λπr Rζ sin t log x+ tθrα L λπr Iζ µg m t dt. B. Aroximations of the Meta Distribution Though the exression in Cor. 3 is exact and can be calculated via numerical integration techniques, it is difficult to gain insights directly and aly it to obtain other analytical results. To aroximate the meta distribution, we first use the same aroach adoted in [7] by matching the mean and variance of the beta distribution with M θ and M θ given in Theorem to verify whether this aroximation is also accetable for mm-wave DD networks, where the interference characteristics are different from those in microwave networks. The robability density function PDF of a beta distributed random variable X with arameters κ and β is f X x = xκ x β, Bκ, β where B, is the beta function. The first and second moment of X are given as EX = κ κ + β, EX = κ + EX. 3 κ + β + Letting EX = M θ and EX = M θ, we have κ = M M M M M, β = M M M M M. 4 Hence, the aroximate meta distribution of the tyical mmwave DD receiver follows as F Ps θx I x κ, β, 5 where I x κ, β is the regularized incomlete beta function. To answer the question when the above aroximation rovides a good match with the exact result in mm-wave networks and when not, we comare them with different numbers of antenna elements N and node densities in Fig. a. It is observed that the standard beta distribution rovides an excellent match for the case with severe interference while as the number of antennas increases with narrower beams or the density of the nodes decreases, the aroximations start to deviate from the exact results. The smaller the interference, the larger the deviations. However, due to the unique features of mm-wave such as high roagation loss, highly directional transmission, and sensitivity to blockage, the interference behavior is different from the microwave communications. The randomness of the interference stems from the satial relative locations between the interferer and the receiver or the interfering beam directions. Due to the randomness of the distribution of the DD users, both strong and weak interference scenarios will occur. Thus, for the sake of aroximating the meta distribution more accurately even for the light interference scenarios in mm-wave bands, we roose another aroximation with a generalized beta distribution whose PDF with arameters κ, β, ρ is given as f X x = xκ x/ρ β ρ κ x ρ, 6 Bκ, β where ρ, ]. Clearly, for ρ =, this reverts to the standard beta aroximation. The arameters can also be obtained through the moment matching method, given as EX = ρκ κ + β, κ + n EXn = ρ κ + β + n EXn, 7

6 6 F x λ=. N=4 λ=. N=64 λ=. N=64 λ=. N=4 F x λ=. N=4 λ=. N=64 λ=. N=64 λ=. N= Beta Arox.. Exact Results Generalized Beta Arox x a SINR analysis. Beta Arox.. Exact Results Generalized Beta Arox x b SIR analysis Fig.. Standard beta aroximation versus generalized beta aroximation with =., r = 3, µ =, =.5, α L =.5, α N = 4. and the solutions to the three equations, i.e., n =,, 3 can be easily obtained via the fsolve function in Matlab 4 and later versions. Thus, the generalized beta aroximation of the meta distribution is obtained by F Psθx I x κ/ρ, β x ρ. 8 From the numerical results in Fig., we find that the generalized beta aroximation rovides an excellent match for the distribution of the link success robability with a wider range of alication. For examle, for N = 64, when a highly directional antenna array is adoted, the generalized beta aroximation rovides a more accurate match than the standard beta distribution, esecially at high reliabilities. Moreover, from λ =. to λ =., when the node density decreases, the deviation of the standard beta distribution becomes quite significant while the generalized beta aroximation still rovides an excellent match. Furthermore, to investigate how the noise affects the meta distribution and its aroximation, we comare the meta distributions of the SINR and the SIR, where the latter is the noiseless case, shown in Fig. b. It is seen that the meta distribution for weak interference is truncated at x < in case a but not in case b, see the two curves with λ =. in each figure. This henomenon indicates that when the node density decreases, the network tends to be noise-limited, and the effect of the noise on the meta distribution cannot be ignored. This is also the reason why ρ < in 6 leads to a better fit for the SINR analysis. Besides, as exlained for Cor., when N becomes larger, the node density becomes the dominant factor of the interference. Thus, when N = 64, due to the higher node density λ =., even though the noise effect is not considered, the meta distribution is also truncated before but now because of the interference. This indicates that for the mm-wave network with large antenna arrays and dense deloyment, the standard beta distribution does not rovide an accurate aroximation of the meta distribution esecially at high reliabilities near, whereas the generalized one does. In summary, by using higher moments of the conditional SINR distribution, the generalized beta aroximation rovides an excellent match for the distribution of the link success robability with a wider range of alication scenarios both interference-limited and noise-limited cases and reverts to the standard one for ρ =. Such a quick and efficient aroximation significantly facilitates the erformance evaluation, which will lay an imortant role for network lanning and management. C. Meta Distribution of the Data Rate In addition to the transmission reliability or the success robability, the data rate, characterized by the rate distribution, is another fundamental erformance metric of transmission effectiveness. Denote T as the random data rate of the tyical link, with unit of bs. According to the Shannon caacity formula T = W log + SINR, we have the rate distribution PT > τ = M τ/w. Similar to the meta distribution of the SINR, conditioned on the oint rocess, we can also derive the meta distribution of the data rate F T τ, x to resent the fraction of active users in each realization of the oint rocess that have a rate T greater than τ with robability at least x. Theorem. Meta distribution of data rate for mmwave DD receivers Given that the tyical link is LOS and active, the meta distribution of the data rate in mm-wave DD networks can be obtained through the moment S b b C of the conditional data rate, where S b τ = M b τ/w. Proof: Since T = W log + SINR, we have PT > τ Φ = PW log + SINR > τ Φ = PSINR > τ/w Φ. 9 Therefore, [ ] b S b τ = E Φ PSINR > τ/w Φ

7 7 = M b τ/w. 3 The first moment of the conditional data rate is the rate distribution for the mm-wave DD network, given as S τ = ex λπr ξ L + N ξ N τrα L, 3 µg m where τ = τ/w, and ξ i = w F, δ i, δ i +, Rαi τr α L + w F, δ i, δ i +, G mr α i G s τr α L, i {L, N}. The exact meta distribution F T τ, x of the data rate can also be calculated via the Gil-Pelaez theorem. Due to the similarity of the meta distribution between the SINR and achievable rate, the beta aroximation also works well in some regimes, and the generalized beta aroximation is accurate in all cases we considered. D. The Mean and Variance of the Local Delay The local delay is another imortant network erformance metric that directly affects the erceived exerience of users. It is defined as the number of transmission attemts needed until the first success [6]. In each transmission attemt, the transmitter is allowed to transmit with robability, and the transmission will be successful with robability P s θ conditioned uon Φ. Therefore, the transmission attemts are indeendent Bernoulli trials with success robability P s θ and the conditional local delay, denoted as D Φ = D Φ, is a random variable with geometric distribution given by PD Φ = k = P s θ k P s θ. 3 Hence the mean conditional local delay is given by ED Φ = P sθ. As a result, the mean local delay D is obtained as D = E P s θ = M, which characterizes the mean number of transmission attemts needed until a acket is successfully transmitted. The secific exression of the mean local delay is given as follows. Theorem 3. Letting v i = Rα i and ṽ i = v ig m G s, the mean local delay D of mm-wave DD networks is D = ex G m µ + λπr i F δ i,,, δ i +, z, z + δ i wṽ i + wv i δ i + F δ i +,,, δ i +, z, z,33 where z and z are the two different real roots of fy = v i ṽ i y + v i + ṽ i wṽ i wv i y +. Proof: See Aendix C. In order to better understand the distribution of the local delay, we also derive its variance to characterize the fluctuation of the delay or jitter. Theorem 4. Letting v i = Rα i and ṽ i = v ig m G s, the variance of the local delay V D in mm-wave DD networks is V D= D ex λ πr i F δ i,,, δ i +, z, z + δ i wṽ i + wv i F δ i +,,, δ i +, z, z δ i + + δ i wṽ i + wv i F δ i +,,, δ i +3, z, z δ i + D D, 34 where z and z are the two different real roots of fy = v i ṽ i y + v i + ṽ i wṽ i wv i y + =. Proof: See Aendix D. E. Satial Outage Caacity The satial outage caacity SOC, introduced in [], is a new notion of caacity that can be calculated using the meta distribution. The SOC is defined as the maximum density of concurrently active links with a success robability greater than a certain threshold. According to the definition, the SOC of mm-wave DD networks is given by Sθ, x su λ F Psθx, θ >, x,. 35 λ>,,] The SOC measures the maximum density of links that satisfy a certain reliability requirement in a mm-wave DD network, where the reliability requirement is alied at each individual link given the SINR threshold θ. We denote the density of concurrently active links that have a success robability greater than x as sθ, x λ F Ps θx. 36 With the accurate aroximation of the generalized beta distribution above, sθ, x is aroximated as sθ, x λ I x κ/ρ, β x ρ. 37 It should be noted that both the meta distribution and the SOC are imortant erformance metrics: the former characterizes a fine-grained link-level erformance of all links i.e., the entire distribution of the random variable instead of just the mean; while the latter characterizes a fine-grained networklevel erformance i.e., the maximum density for a network with a certain QoS constraint alied at each individual link instead of the standard area sectral efficiency ASE which is just the mean achievable rate er unit area given that the tyical link satisfies the QoS constraint. Thus, from these two erformance metrics, we cature the erformance of individual links and obtain much sharer results than merely the SINR and ASE at the tyical user. IV. NUMERICAL RESULTS In this section, we will resent numerical results of various erformance metrics involved in the framework in Section III for mm-wave DD networks. The main symbols and arameters are summarized in Table II with default values in the simulations.

8 8 TABLE II. Symbols and descritions Symbol Descrition Default value Φ, λ mm-wave device PPP and density N/A,./m µ The transmit ower db W mm-wave bandwidth GHz r The link distance between the DD users 3m The transmit robability in each time slot.5 / N The robability of a link being LOS/NLOS./.8 α L /α N The ath loss exonent of the LOS/NLOS link.5/4 R The radius of the generalized LOS ball m θ The SINR threshold db G m/g s The main lobe/side lobe of the antenna attern N/A w The beam-width of the antenna attern N/A F Ps θx/ F T x, τ The meta distribution of the SINR/data rate N/A M b /Mb τ The b-th moments of the conditional SINR/data rate distribution N/A Standard Success Probability N = 4 N = 6 N = 64 =. =.5 N Variance N = 4 N = 6 N = 64 =. =.5 N 3 SINR Threshold db 3 SINR Threshold db Fig.. The standard success robability M. Fig. 3. The variance M M of the conditional success robability. A. The Imact of mm-wave Features Directional transmission and sensitivity to blockage are two key features of mm-wave communication. In this subsection, we focus on the imacts of the LOS robability and antenna array size N on the mean and variance of the conditional success robability P s θ. Moreover, we also investigate the interference behaviors under various mm-wave DD scenarios. As shown in Fig., increasing the number of antennas imroves the standard success robability due to the fact that a larger antenna array can form a narrower beam, thus causing less interference. As the number of antennas tends to infinity, the standard success robability converges to an uer limit see Cor., since the side-lobe leakage restricts the erformance imrovement. However, for the LOS robability, a higher means the number of LOS links is larger, leading to more severe interference. Thus, the standard success robability deteriorates with the increase of. Fig. 3 resents the variance of the conditional success robability as a function of θ for different LOS robabilities and antenna array sizes. Since the variance necessarily tends to zero for both θ and θ, it assumes a maximum at some finite value of θ. It can be seen that given a, the θ with the maximum variance increases with N and converges to an uer limit, similar to M. Moreover, there is no monotonicity of the variance with resect to N. For examle, for θ = db, the variance decreases with increasing N, while the oosite haens for θ = db. We also find that a larger leads to smaller variance. The reason is that in LOS environment with smaller ath loss exonent, the randomness of the relative distances of the interferers and the receiver has a smaller effect on the variance of the interference than in the NLOS case. Fig. 4 shows the relationshi between the standard success robability and node density for θ = db. In order to find whether and when mm-wave DD networks are noiselimited, we also lot a SNR standard success robability where interference is neglected. As seen from the lots, the mmwave network tends to be interference-limited as the density increases and for the given system arameters, there is a critical oint λ c 4. The critical density λ c is the density where the interference starts to be the dominant term in the interference-lus-noise sum, i.e., when PI > is high recall that the noise ower is set to. Therefore, when λ > λ c, the noise-limited assumtion is no longer validated.

9 9.9.8 r =4.9.8 Beta Arox. w. λ=. & =.5 Beta Arox. w. λ=.5 & =. Beta Arox. w. λ= & =.5 Exact Results Standard Success Probability r = SINR SNR =,.5,.5 =,.5,.5 F x λ x Fig. 4. Standard success robability versus density for N = 4. Fig. 5. Meta distribution for different λ and with N = 4, =...9 θ= db.9 Beta Arox. Exact Results.8.8 N=64, =. F x θ=5db F x N=4, =. N=6, =..3.3 N=4, =... Beta Arox. Exact.. N=4, = x x Fig. 6. Meta distribution for θ =, 5,, 5,, 5 db. Fig. 7. Meta distribution for different and N. Moreover, the higher the density, the larger the deviations between the SINR and SNR erformance. It is well known that DD communication mostly alies in crowded environments such that users in close roximity can establish reliable direct communications. Therefore, it can be concluded that the mmwave DD network is interference-limited in most cases. Besides, the selection of the link distance between the DD users is also imortant since the interferers can be arbitrarily close to the receivers; if the link distance is too large e.g., the case r =, the standard success robability will be seriously deteriorated. B. Meta Distribution in mm-wave Band Fig. 5 shows comarisons of the exact results and beta aroximations for λ =.5 with different λ and, resectively. As seen from the lots, for the given system arameters, the aroximations match the exact results extremely well, which verifies the accuracy and the effectiveness of the aroximation. Moreover, the three curves have the same value of λ and hence the same standard success robability, but the corresonding meta distributions are rather different. This shows that the standard success robability rovides only limited information on the network erformance. Fig. 6 shows the meta distributions for different SINR thresholds, which enables a recise statement about what fraction of links achieve an SINR threshold with a target reliability. For examle, for θ = 5 db, about 9% of the links have a success robability of at least 8%; while for θ = 5 db, less than % of the links achieve the same reliability. Fig. 7 shows the imacts of LOS robability and antenna array size on the meta distribution. As seen from the lots, adoting a large antenna array boosts the erformance. For examle, the fractions of links with a success robability of at least 6% for N = 6 and N = 64 are about.7 and.8, resectively, which are significantly higher than for N = 4 where almost no links meet the requirement. In addition, similar to the standard success robability, the increase of leads to more interferers with LOS links, thus resulting in oor erformance. Fig. 8 and Fig. 9 illustrate the rate distribution and the

10 .9.9 N=4 & =. L N=6 & = N=64 & =. N=4 & =.5 L N=4 & =. L CDF N = 4 N = 6 N = 64 =. F T x = Achievable rate Gbs Fig. 8. The rate distribution for different N and x Fig. 9. Meta distribution of the achievable rate for different N and with rate threshold τ = Gbs. F x Rayleigh w. N=64, λ=. Nakagami w. N=64, λ=. Rayleigh w. N=4, λ=. Nakagami w. N=4, λ=. Rayleigh w. N=64, λ= Nakagami w. N=64, λ= x Fig.. Comarisons of meta distribution between Rayleigh and Nakagami-4 fading. meta distribution of the conditional achievable rate with rate threshold τ = Gbs, resectively. Due to the rich bandwidth resource, for the given system arameters, mm-wave DD communication can rovide multi-gigabit-er-second rates even at moderate SINRs. This observation demonstrates the unique advantages of mm-wave communications esecially for mobile broadband services. As a result, the load of the base stations and the backhaul networks could be significantly reduced. From both figures, increasing the size of the antenna array has a ositive effect on the achievable rate and its meta distribution, i.e., increasing both the mean achievable rate and the fraction of concurrent links achieving a required rate. The LOS robability can also affect the ath loss law of the interfering links, and thus a smaller robability leads to reduced interference. These observations hel network oerators find the most efficient oerating regime for DD communication in the mm-wave band. To assess the imact of the fading model, we give a comarison of two cases: one adots the Nakagami-m fading for LOS roagation with m = 4; and the other adots the Rayleigh fading, shown in Fig.. It is shown that the Nakagami and Rayleigh fading resent the same trends in terms of the meta distribution under different cases of densities and numbers of antennas, which imlies the significance of the theoretical results based on the Rayleigh fading. C. Mean and Variance of the Local Delay Fig. and Fig. give the mean and variance of the local delay for different N and, resectively. It can be observed that both statistics of the local delay aear to resent similar trends with the transmit robability. A small means less oortunity to be scheduled and thus lengthens the transmission eriod, while a large results in severe interference and thus reduces the successful transmission robability. Thus, both of the two cases increase the delay and jitter. Also, the otimal that minimizes the mean local delay does not corresond to the smallest delay jitter, which highlights the imortance of a fine-grained analysis for the delay erformance, esecially for the real-time alications which are sensitive to the delay jitter. Moreover, the unique features of mm-wave communication, e.g., the antenna array size and the LOS robability, also have significant effects on the delay erformance. As seen from the figure, a larger antenna array and a smaller LOS robability reduce the delay while maintaining the corresonding delay stability in a large range of the transmit robability, which verifies the advantages of mm-wave communications in terms of the delay erformance. D. SOC analysis Fig. 3 exlores the behavior of sθ, x for fixed θ = db and x =.9 as a function of and λ for different N and and indicates the SOC oint which is the combination of λ, achieving the suremum of sθ, x. It is observed that for an arbitrary combination of N and, the SOC is always achieved at = under the given system arameters. Besides,

11 5 Mean Local Delay N=4 N=6 N=64 =.5 =. The Variace of Local Delay 5 N=4 N=6 =.5 =. N= Transmit Probability Fig.. The mean local delay with different N and Transmit Probability Fig.. The variance of the local delay with different N and a N 4,. L.9 =.4 = S=.35 SOC oint = = S=.6 SOC oint b N 6,. L =.5 = S= SOC oint c N 64,. L =. = S=.46 SOC oint =.7 = S=.468 SOC oint.6.4 = = S=.675 SOC oint d N 4, L 5. e N 6, L 5. f N 64, L 5. Fig. 3. Contour lots of sθ = /, x =.9 as a function of and λ for different N and given λ or, sθ, x tends to increase first to reach a critical oint and then decreases as or λ increases. This is because the increase of or λ leads to two oosite effects on sθ, x, namely an increase in the concurrently transmitting links λ and a decrease in the fraction F Psθx of reliable links due to the increase of interference. Therefore, there exists a maximum as or λ increases. It is also observed that larger N and smaller lead to larger SOC, which can be interreted as follows: a larger N generates narrower transmitting beams concentrating more ower to the dedicated receiver and thus causing less interference to other concurrent links; a smaller makes more interfering links to be NLOS, and thus the interference suffered by the receiver is also reduced. Both oerations increases the fraction of successful transmission links, and thereby making the networks be able to suort more concurrent links, i.e., a larger SOC can be obtained. As a baseline, the SOC serves a critical role that informs the network oerator whether the mm-wave DD network satisfies the caacity requirement.

12 V. CONCLUSIONS In this aer, we roose a general framework for a fine-grained analysis of mm-wave DD networks based on the meta distribution. We first derive the moments on the conditional success robability P s for DD receivers and then rovide an exact exression as well as a simle yet accurate aroximation for the meta distribution of the SINR. These results are then extended to the meta distribution of the achievable rate and alied to the mean and variance of the local delay as well as the SOC. Hence, the comlete distribution of both the conditional link success robability and the conditional link data rate in mm-wave DD networks can be characterized, which rovides much sharer results than merely the means i.e., the standard success robability and the mean achievable rate. Also, the SOC gives a networklevel erformance metric based on the meta distribution with a certain reliability constraint or QoS requirement. Using this framework, we fully exlore the imacts of the unique features of the mm-wave band on the erformance of DD networks through both theoretical and numerical studies and obtain the following useful insights: the concentration result, obtained in biolar networks with standard ath loss and single antenna, still holds in mm-wave DD networks, i.e., the variance of P s goes to as the transmit robability while keeing the mean success robability constant; the standard success robability increases monotonically with the antenna array size N and converges to a maximum as N, where the noise is totally suressed and the side lobe interference as well as the node density become the dominant factors in determining the interference; 3 the sensitivity to blockage is another imortant feature of the mm-wave band that affects the interference characteristics directly and hence the erformance metrics such as the success robability, the local delay and the SOC. In summary, the mm-wave DD technique is exected to bring huge benefits for future wireless networks. However, the salient roerties of mm-wave should be carefully exlored in order to exloit them for DD communication. ACKNOWLEDGMENT The helful comments by Dr. Sanket Kalamkar are gratefully acknowledged. APPENDIX A PROOF OF THEOREM Proof: Letting θ = /G m, we have from 5 P s θ = E ex θ I + /µ Φ =e θ µ E + θ gφ x lx + x Φ =e θ w w µ +θ G m x α + i +θ G s x α +, i x Φ i Letting δ i = /α i, i {L, N}, we have M b =E [ P s θ b] =e bθ µ E Φi x Φ i w + θ G m x α i a R = e bθ µ w ex πλ i + θ G m r α i = e bθ µ λπr ex w + yrα i θ G m b = e bθ µ ex b w + + λπr i δ i + θ G s x α i + w b + θ G s + rdr r α i b w y dy δi + yrα i θ G s b λπr i δ i n+ n n n= n n w m w n m y δi dy m m n m m= + yrα i + yg mr α i Gs c = e b Gmµ ex b n n λπr i n+ n w m n m n= w n m F δ i, m, n m, δ i +, Rα i m=, G mr α i G s, where ste a uses the robability generating functional PGFL of the PPP [7], ste b follows from the general binomial theorem and ste c is obtained with the hel of the formula in [4, Eq. 3.]. APPENDIX B PROOF OF COROLLARY Proof: According to the roof of Theorem, the moment M b is given by M b = e b Gmµ λπr ex λπr i δ i w + yrα i w + yrα i G m G s b y δ i dy,38 where G m = N, G s = / sin 3π and w = 3 N π, N > N. It should be noted that N = is the omnidirectional case and thus w =. First of all, when b R +, we know that the term e bθrα L /G mµ λπr is monotonically increasing with N. Hence, letting x =, whether M b is monotonically yrα i increasing with N deends on whether fn = w w +x + + xg m Gs is monotonically decreasing with N. In the following, we rove that fn = fn decreases monotonically, given by 3 fn= πn+x + πn 3 πn +xn sin, N, 39 3π N and f = / + x, where fn and fn have the same monotonicity.

13 3 Secondly, we rove fn is monotonically decreasing with N. Relaxing to N [,, we obtain the first-order derivative of fn, given as 3 f N = πn +x +xn sin 3π N xπn 3 sin 3π N χn πn + xn sin 3π, 4 N where χn = 4N + 9π sin 3π N φ and tanφ = 3π N. Since N, we have N sin 3π N > and χn >, thus f N < and fn is monotonically decreasing with N. Thirdly, we can easily rove f f = 3 +x 4π+x + 4π 3 4π +x4 sin 3π 4 >.4 In summary, fn is monotonically decreasing with N and so is fn. Thus, M b is monotonically increasing with N. When N, we have w, G m and G m /G s 9π /4. As a result, we have lim M b N =e λπr ex λπr i δ i + yrα i 9π 4 a =ex b λπr i δ i n+ n n n= b =ex b λπr i n+ n n n= b y δi dy y δi dy n + y9π R α i 4 F n, δ i, δ+, 9π R α i, 4 4θr αl where ste a uses the general binomial theorem and ste b follows from the definition of the Gaussian hyergeometric function in [4, Cha. 9.]. APPENDIX C PROOF OF THEOREM 3 Proof: The mean conditional local delay is given by ED Φ = P s θ, and by averaging over Φ, the mean local delay follows as [ ] D = E P s θ = M θ = ex G m µ λπr + λπr i δ i w w + yrα i + yrα i G y δi dy.43 m }{{ Gs } A i Letting v i = Rα i and ṽ i = v ig m G s, we have + v i y + ṽ i yy δ i A i = δ i +v i y+ṽ i y + wṽ i + wv i y dy + wṽ i + wv i yy δi dy =+δ i +v i y+ṽ i y + wṽ i + wv i y.44 It is easy to rove that the equation of the denominator in 44 has exactly two different real roots. Denoting them by z and z, we have A i = + δ i + wṽ i + wv i yy δ i dy z y z y = + F δ i,,, δ i +, z, z + δ i wṽ i + wv i F δ i +,,, δ i +, z, z. 45 δ i + Inserting 45 in 43, we obtain 33. APPENDIX D PROOF OF THEOREM 4 Proof: The variance of the local delay is given as V D = ED D = EEDΦ D a Ps θ = E P s θ D = M θ D D, 46 where ste a follows from the second moment of the geometrically distributed random variable, and M θ is given by [ ] M θ = E P s θ θr αl = ex G m µ λπr + λπr i δ i w w + yrα i + yrα i G y δi dy.47 m G s }{{} Similar to the roof in the Aendix C, we have + wṽ i + wv i y y δi B i = +A i +δ i z y z y dy = + A i + F δi,,, δ i +, z, z + δ i wṽ i + wv i F δ i +,,, δ i +, z, z δ i + + δ i wṽ i + wv i F δ i +,,, δ i + 3, z, z. δ i + B i Inserting 47 in 46, we obtain 34.